Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Semiclassical analysis for highly degenerate potentials

Authors: P. Álvarez-Caudevilla and J. López-Gómez
Journal: Proc. Amer. Math. Soc. 136 (2008), 665-675
MSC (2000): Primary 35B25, 35P15, 35J10, 31C12
Published electronically: November 2, 2007
MathSciNet review: 2358508
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Abstract: This paper characterizes the semi-classical limit of the fundamental energy,

$\displaystyle E(h):= \sigma_1[-h^2\Delta+a(x);\Omega], $

and ground state $ \psi_h$ of the Schrödinger operator $ -h^2\Delta+a$ in a bounded domain $ \Omega$, in the highly degenerate case when $ a\geq 0$ and $ a^{-1}(0)$ consists of two components, say $ \Omega_{0,1}$ and $ \Omega_{0,2}$. The main result establishes that

$\displaystyle \lim_{h\downarrow 0} \frac{E(h)}{h^2}= \min\left\{\sigma_1[-\Delta;\Omega_{0,i}], \; i=1,2\,\right\} $

and that $ \psi_h$ approximates in $ H_0^1(\Omega)$ the ground state of $ -\Delta$ in $ \Omega_{0,i}$ if

$\displaystyle \sigma_1[-\Delta;\Omega_{0,i}]< \sigma_1[-\Delta;\Omega_{0,j}],\qquad j \in \{1,2\}\setminus\{i\}. $

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Additional Information

P. Álvarez-Caudevilla
Affiliation: Departamento de Matemáticas, Universidad Católica de Ávila, Ávila, Spain

J. López-Gómez
Affiliation: Departamento de Matemática Aplicada, Universidad Complutense de Madrid, 28040-Madrid, Spain

Keywords: Fundamental energy, ground state, highly degenerate potentials, classical conjecture of B. Simon, compact Riemann manifolds.
Received by editor(s): January 19, 2007
Published electronically: November 2, 2007
Additional Notes: This work was partially supported by the Ministry of Education and Science of Spain under research grants REN2003–00707 and CGL2006-00524/BOS
Communicated by: David S. Tartakoff
Article copyright: © Copyright 2007 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.